I would like to construct a real-valued function $f$ on $(0, \infty)$ with the following properties:
$f(r)$ is $C^1$ on $(0,\infty)$ and $C^\infty$ on $(0,1) \cup (1, \infty)$,
$-f'' + \tfrac{3}{4}r^{-2}f = (C - 1)f$ on $(0,1)$, for some $C >0$,
$-f'' + \tfrac{3}{4}r^{-2}f = -f$ on $(1,\infty)$
$f(r) > 0$ for all $r > 0$,
$f(r)$ decays exponentially toward $\infty$,
$f(r) \sim r^{3/2}$ as $r \to 0$.
Such an $f$ is used in Simon 1973 (Essential Self-Adjointness of Schrödinger operators with singular potentials) to establish essential self-adjointness on $C^\infty_0(\mathbb{R}^n \setminus \{0\})$ for a certain class of Schrödinger operators with potentials singular at zero.
For any $C > 0$, the function $$f_{< 1}(r) = \frac{2}{\sqrt{C-1}} r^{1/2} J_1(\sqrt{C-1}r)$$ is a solution to the first ODE with the desired properties on $(0,1)$. Here, $J_1$ is the Bessel function of the first kind of order one, and we take $\sqrt{C-1}$ to be on the positive imaginary axis if $C - 1 < 0$.
On the other hand, the function $$f_{>1}(r) = - r^{1/2} H^{(1)}_1(ir)$$ is a solution to the second ODE with the desired properties on $(1,\infty)$. Here, $H^{(1)}_1$ is the Hankel function of the first kind of order one.
With these two solutions in hand, Simon claims that by adjusting the positive constant $C$, "we can arrange that the two solutions and their derivatives match at $r = 1$" (ensuring that $f$ is $C^1$ as desired). It is this final claim that I have trouble understanding. I don't see how being able to choose $C > 0$ grants enough degrees of freedom to match the values of $f_{< 1}$ and $f_{> 1}$ and their derivatives at $r =1$. Perhaps one needs to define $f_{< 1}$ and $f_{> 1}$ differently from what I have proposed?
